Liquid crystal based polarimetric system, a process for the calibration of this polarimetric system, and a polarimetric measurement process

ABSTRACT

A liquid crystal based polarimetric system, a process for the calibration of this polarimetric system, and a polarimetric measurement process intended for measuring the representative parameters of a sample in which the polarimetric system contains an excitation section emitting a light beam that passes through a polarization state generator (PSG) and onto a sample. After reflection or transmission by the sample, the beam goes through an analysis section with a polarization state detector (PSD). The PSG and PSD each have a first and a second liquid crystal elements LC j  (j=1,2) having, for each LC j  element of the PSG (respectively for each LC j  element of the PSD), an extraordinary axis making an angle θ j  (resp. θ′ j ) with respect to the polarization direction (i), and a retardation δ j  (resp (δ′ j ) between its ordinary and extraordinary axes, the liquid crystals LC j  elements being positioned in reverse order in the PSD with respect to the LC j  elements of the PSG.

[0001] The invention relates to a liquid crystal based polarimetricsystem for analyzing a sample, a process for the calibration of thispolarimetric system and a polarimetric measurement process.

[0002] In order to measure parameters which are representative of asample (for example, of its composition and thickness), it isconventional to make use of an ellipsometer.

[0003] Ellipsometry is a powerful non invasive probe in whichreflectance or transmittance data are measured by electromagneticradiation outcoming from a sample. Briefly, the surface of a studiedsample is illuminated by a photon beam that is reflected or transmittedand the polarization state of the outcoming beam is compared to that ofthe incident beam.

[0004] This conventional ellipsometry method proves satisfactory whenthe reflected light is still totally polarized (even thoughelliptically), which is indeed the case, among other examples, forisotropic layers with smooth interfaces. Such samples, which can beconsidered as “dichroïc retarders” (DR) are usually characterized by theellipsometric angles (Ψ,Δ) defined by

r _(p) /r _(s)=tan(Ψ)exp(iΔ)  (1)

[0005] where r_(p) and r_(s) are respectively the amplitudereflectivities of the sample for linear polarizations in the incidenceplane (r_(p)), or perpendicular to this plane (r_(s)).

[0006] However, to study samples that cannot be described as DRs, suchas partially depolarizing materials, a more general method is required.

[0007] Polarimetric systems enable to measure all the polarizationcomponents of light in any sample.

[0008] The polarization state of light can be represented in the mostgeneral case by a four-dimensional vector, called the Stokes vector (S).

[0009] A description of this can be found in the work of Azzam andBashara entitled “Ellipsometry and polarized light”, North-Holland, pp.55-60.

[0010] The Stokes vector consists of the I, Q, U and V coordinates. Istands for the total intensity, while the other components are definedas the differences between the intensities measured through differentpairs of “complementary” polarizers (linear polarizers set vertical andhorizontal for Q, at +45° and −45° azimuthal angles for U, and left andright circular polarizers for V).

[0011] The interaction of light with any sample can then be representedby a matrix, so-called, the Mueller matrix, of dimensions 4×4 withtherefore 16 real coefficients.

[0012] The extraction of the 16 parameters during polarimetricmeasurements provides a complete characterization of the said medium.

[0013] For a DR characterized by ellipsometric angles (Ψ, Δ) (see eq.(1)) the Mueller matrix elements are the following:

[0014] upper diagonal block

M ₁₁ =M ₂₂ =τ, M ₁₂ =M ₂₁=−τ cos(2Ψ)

[0015] lower diagonal block

M ₃₃ =M ₄₄=τ sin(2Ψ)cos(Δ), M ₃₄ =−M ₄₃=τ sin(2Ψ)cos(Δ)  (2)

[0016] other elements: M_(ij)=0

[0017] where the additional parameter τ is proportional to the overallintensity transmission or reflection coefficient of the sample. We pointout that this Mueller matrix has

[0018] two real eigenvalues: (λ_(R1)=2τ sin²(Ψ), λ_(R2)=2τ cos²(Ψ)),

[0019] and two complex conjugate eigenvalues:

(λ_(C1)=τ sin(2Ψ)exp(iΔ), λ_(C2)=τ sin(2Ψ)exp(−iΔ)).  (3)

[0020] Many designs have been proposed, and demonstrated, for Muellerpolarimetric systems. All of them comprise a polarization stategenerator (PSG) which modulates the Stokes vector (S_(in)) of the lightimpinging on the sample and a polarization state detector (PSD) whichanalyzes the polarization (S_(out)) of the light outcoming from thesample. It is customary to define the modulation matrix W as a 4×4 realmatrix whose columns are the four Stokes vectors S_(in) generated by thePSG. Reciprocally, the four dimensional signal vector D eventuallydelivered by the PSD is related to the Stokes vector S_(out) of thelight outcoming from the sample by a linear relationship D=A S_(out),where A is the (4×4 real) analysis matrix representing the PSD. A rawmeasurement actually consists of 16 values of the signal, which form amatrix B=A M W, where A and W are respectively the analysis andmodulation matrices defined above, and M the Mueller matrix of thesample. If A and W are known, M can be extracted from the raw data B asM=A⁻¹ B W⁻¹. The determination of A and W is called calibration of thepolarimeter. Clearly, the instrument must be designed in such a way thatA and W are non singular. Moreover, in order to optimize errorpropagation from the raw measurement B to the final result M, theanalysis and modulation matrices A and W must be “as close as possible”to unitary matrices. The best criterion in this respect is to optimizetheir condition numbers s(A) and s(W), which are the ratios of thesmallest over the largest of their singular values {see for examplePress W. H., Teukolsky S. A., Vetterling W. T. and Flannery B. P.;Numerical Recipes in Fortran, Cambridge University Press, p 53, whoactually define the condition number as the reciprocal of that definedhere; see also Compain E. and Drévillon B.; Rev. Sci. Instrum. 69,(1998) 1574}.

[0021] In a PSG, the light polarization can be modulated by a variety ofdevices such as discrete components inserted and then removed from thelight path {Bickel W. S. et al.; Am. J. Phys 53 (1984) 468}, rotatingretardation plates {Goldstein D. H.; Appl. Opt. 31 (1992) 6676},rotating compensators {Collins R. W. and Koh J.; J. Opt. Soc. A 16,(1999) 1997}, Pockels cells {Delplancke F.; Appl. Opt. 36 (1997) 5388and Compain E. and Drévillon B.; Rev. Sci. Instrum. 68 (1997) 2671} orphotoacoustic modulators {Compain E. and Drévillon B.; Rev. Sci.Instrum. 69, (1998) 1574}. For PSD, one can use the same devices and asingle detector, or a “parallel” analysis of light polarization throughpolarization-sensitive beamsplitters and simultaneous measurement of theseparated beams by several detectors {Azzam R. M. A., Opt. Acta 29(1982) 685, Brudzewski K.; J. Modern Optics 38 (1991) 889, Khrishnan S.;J. Opt. Soc. Am A 9 (1992) 1615, Compain E. et al., Thin Solid Films 313(1998)}.

[0022] This variety of designs leads to a variety of characteristics,some of which are not compatible with each other; for example, highfrequency polarization modulation by resonant devices such asphotoelastic modulators allows efficient rejection of low-frequencynoise, but for imaging applications by slow detectors such as typicalCharge Coupled Devices (CCD), a stroboscopic illumination is thenneeded. Another important requirement for imaging applications is thatthe polarimetric elements exhibit large enough acceptance anglestogether with small enough aberrations. As a result, the devices whichbest meet these requirements are those based on low order retardingplates, which are either rotated {Pezzaniti J. L. and Chipman R. A.,SPIE proceedings 2297 (1994) 468} or inserted and removed betweensequential measurements, or those based on liquid crystal (LC) variableretarders.

[0023] Liquid crystal cells (LC) are electrically controlled low orderretardation plates. Two types of such devices are currently available.First, nematic liquid crystals (NLC) provide variable retardation withfixed orientation of slow and fast axes, with typical response times ofthe order of 10 to 100 ms. On the other hand ferroelectric liquidcrystals (FLC) provide fixed retardation, but with slow and fast axisdirections which can be electrically switched between two azimuthalangles separated by 45°, in times typically shorter than 100 μs.

[0024] These elements provide non resonant polarization control which isnaturally well suited for polarimetric imaging by a CCD. Therefore,devices comprising liquid crystal cells have been proposed forpolarimetric imaging within the frame of conventional ellipsometry, i.e.for samples behaving as DR {Oldenbourg R. et al.; J. Microscopy 180(1995) 140} and led to commercially available devices (Pol-Scole, byCRI, Inc. Boston).

[0025] Stokes polarimetry, i.e. polarimetry performed using a solepolarization state detector and no polarization state generator has alsobeen performed, essentially in solar astronomy. The device consisted oftwo nematic LCs followed by a linear polarizer {Hofmann A.; SPIEproceedings 4133 (2000) 44} or even more complex assemblies, includingfor example two ferroelectric LCs, two fixed λ/8 retardation plates anda linear polarizer {Gandorfer A. M.; Opt. Engineering 38 (1999) 1402} orone ferroelectric, two nematic LCs and two quarter wave retardationplates {November L. J. and Wilkins L. M.; SPIE proceedings 2265, 210}.

[0026] An imaging Mueller polarimeter has been realized by using nematicLC cells {Bueno J. M.; J. Opt. A: Pure and Applied Optics 2 (2000) 216}.In this device, the PSG and PSD have the same design: each of themconsists of one LC and one quarter-wave plate, the latter beingmechanically inserted in and removed from the light path betweenacquisitions of raw images. This device has been used for polarimetricimaging of human eye, including retina and cornea.

[0027] However, the Mueller polarimeters described above, suitable forpolarimetric imaging by slow devices such as CCDs, present twosignificant shortcomings.

[0028] First, their operation involves mechanical motion of opticalelements, which are either rotated or moved in and out of the lightpath.

[0029] Second, their calibration relies on the characterization ofindividual optical components (polarizers and retardation plates). As aresult, the accuracy of the overall calibration of the polarimeter islimited by the accumulation of the errors on the knowledge of each ofthose components, and on their positions. Furthermore, the instrumentalconfigurations are most frequently defined in such a way that thepolarization states generated by the PSG, or those “filtered” by the PSDare “simple” polarization states, such as linear (vertical andhorizontal) and circular states, to reduce “cross-talk” and facilitatethe overall calibration of the system. These “simple” configurations arefar from those providing the highest values of s(A) and s(W), implyingthat for a given input noise on the raw data (B), the noise on the finalresult (M), is far from being optimized.

[0030] A purpose of the present invention is to remedy the shortcomingsmentioned above and to propose a polarimetric system having one or moreof the following features and advantages: namely, polarizationmodulation by liquid crystals only, with no mechanically moving parts,providing a wide enough acceptance angle for typical imagingapplications, a simple and compact design including optimization withrespect to error propagation, and a fast data acquisition, to be usablefor measurements in real time.

[0031] Another purpose of the present invention is to provide twomeasurement processes, which can be used on the same instrument, andwhich yield:

[0032] In a simplified operation mode: the classical ellipsometricangles (Ψ, Δ) of a sample assumed to be a DR. In addition, the validityof this assumption (which depends on the sample homogeneity, roughness .. . ) is tested automatically with no extra measurement, while withusual ellipsometer said test requires a mechanical rotation of theinstrument output arm,

[0033] In a complete operation mode: the complete Mueller matrix (M) ofany sample under study, either in transmission or in reflection.

[0034] The optimization of the PSG and PSD configurations with respectto error propagation implies that the states generated by the PSG (andthose “filtered” by the PSD) are not “simple” ones, such as linear orcircular, and thus usual calibration methods are not really adequate.The invention includes therefore an objective calibration process foreach of the two types of measurement cited above, said calibrationprocesses being at once accurate, rapid and easy to implement.

[0035] To this end, the invention concerns a polarimetric system foranalyzing a sample comprising

[0036] an excitation section emitting a light beam, said excitationsection comprising a polarization state generator containing a polarizerlinearly polarizing the incident light beam along a direction ofpolarization (i),

[0037] an analysis section comprising a polarization state detectorcontaining an analyzer, and detection means,

[0038] a processing unit.

[0039] According to the invention,

[0040] the polarization state generator (PSG) comprise a first and asecond liquid crystal elements LC_(j) (j=1, 2) having, for each LC_(j)element of the PSG an extraordinary axis making an angle θ_(j) withrespect to the direction of polarization (i) and a retardation δ_(j)between its ordinary and extraordinary axes. Said liquid crystal (LCs)elements are placed after the polarizer and are equivalent toelectrically controlled retardation plates,

[0041] the polarization state detector (PSD) comprise a first and asecond liquid crystal elements LC_(j) (j=1, 2) having, for each LC_(j)element of the PSD an extraordinary axis making an angle θ′_(j) withrespect to the direction of polarization (i) and a retardation δ′_(j)between its ordinary and extraordinary axes. Said liquid crystal LC_(j)elements equivalent to electrically controlled retardation plates arepositioned in reverse order in the PSD with respect to the LC_(j)elements of the PSG.

[0042] According to various embodiments, the present invention alsoconcerns the characteristics below, considered individually or in alltheir technical possible combinations.

[0043] In a first embodiment, the polarization state generator (PSG) andthe polarization state detector (PSD) comprise each a first and a secondnematic liquid crystal elements NLC_(j) (j=1, 2). For each NLC_(j)element of the PSG (respectively, for each NLC_(j) element of the PSD),the extraordinary axis makes a fixed azimuthal angle θ_(j) (resp.θ′_(j)) with respect to the direction of the input polarizer of the PSG,(resp. the output analyzer of the PSD). The retardation δ_(j) (resp.δ′_(j)) between NLC_(i) ordinary and extraordinary axes is varied bymeans of an electrical control,

[0044] The azimuthal angles θ′_(j) are equal to θ_(j) (j=1, 2) and theretardations δ′_(j) are equal to −δ_(j) (j=1, 2) (modulo 2π) forsimultaneous optimization of the condition numbers s(W) and s(A) of themodulation and analysis matrices.

[0045] By means of proper driving voltages, the couple of retardations(δ₁, δ₂) takes sequentially the following values: (Δ₁,Δ₁), (Δ₁,Δ₂),(Δ₂,Δ₁), (Δ₂,Δ₂), where Δ₁ and Δ₂ verify the formulae (315°+p 90°) and(135°+p 90°) respectively, where p is the same integer in both formulae,and

[0046] The angles θ₁ and θ₂ verify the formulae (ε 27°+q 90°) and (ε72°+r 90°) respectively where ε=±1 has the same value in both equationswhile q and r are any integer, with tolerances on the angles θ_(i) andon the retardations Δ_(i) equal to +−10° and +−20° respectively. Withsuch tolerances, the condition numbers s(A) and s(W) are then alwaysbetween the maximum value (equal to$( {{{equal}\quad {to}\quad \frac{1}{\sqrt{3}}} \approx {0,58}} )$

[0047]  ) and 0,3 which implies that the noise in the final matrix M(which is inversely proportional to s(A) and s(W) for a given noise onthe raw data B), never exceeds twice its minimum value.

[0048] For spectroscopic applications, i.e. for operation at a variablewavelength, a monochromator is placed before the polarizer of the PSG,or after the analyzer of the PSD, and the values of the retardations(δ₁, δ₂) are kept within the boundaries specified above by simply tuningthe amplitudes of the control voltages according to the wavelengthpassing through the monochromator. The currently available NLCs can beused from 400 nm to 1500 nm typically.

[0049] In an alternative embodiment, the polarization state generator(PSG) and the polarization state detector (PSD) comprise each a firstand a second ferroelectric liquid crystal elements FLC_(j) (j=1,2). Foreach FLC_(j) element of the PSG (respectively, for each FLC_(j) elementof the PSD), the retardation δ_(j) (resp. δ′_(j)) between FLC_(i)ordinary and extraordinary axes is now fixed. For a given set of drivingvoltages the extraordinary axes of the FLC make a couple of azimuthalangles (θ₁,θ₂) (resp. (θ′₁,θ′₂)) with respect to the direction of theinput polarizer of the PSG (resp. the output analyzer of the PSD), andthen these angles are set sequentially to (θ₁,θ₂), (θ₁+45°,θ₂),(θ₁,θ₂+45°), (θ₁+45°,θ₂+45°),

[0050] The azimuthal angles θ′_(j) are equal to θ_(j) (j=1, 2) and theretardations δ′_(j) are equal to −δ_(j) (j=1, 2) (modulo 2π) forsimultaneous optimization of the condition numbers s(W) and s(A) of themodulation and analysis matrices.

[0051] The retardations (δ₁, δ₂) are given by δ₁=80°+−15° andδ₂=160°+−15°, while the orientation angles (θ₁, θ₂) are given byθ₁=67°+−10° and θ₂=160°+−40°. With these values and tolerances, thecondition numbers s(A) and s(W) are again between the maximum value(equal to$( {{{equal}\quad {to}\quad \frac{1}{\sqrt{3}}} \approx {0,58}} )$

[0052]  ) and 0,3.

[0053] For spectroscopic applications, i.e. for operation at variablewavelengths, as the values of the retardations (δ₁, δ₂) are notelectrically controllable as for the nematic crystal elements, thecondition numbers s(W) and s(A) cannot be kept above 0.3 throughout thevisible with a PSG (or a PSD) comprising a linear polarizer and two FLCsonly. However, with typical birefringence dispersion of ferroelectricliquid crystals such a broadband optimization of the condition numbers(s(W) and s(A)>0.3 in the whole spectrum covered by currently availableFLCs (420-800 nm typically) is achieved by adding to the system anotherbirefringent fixed element (retardation plate). An example of such anoptimization is described hereafter, with a quartz plate insertedbetween the two FLCs. The broadband ferroelectric-based PSG thencomprises:

[0054] a linear polarizer, set at an orientation angle θ=0,

[0055] a first ferroelectric liquid crystal, with a retardationδ₁=90°+−5° at 510 nm, set at an orientation angle θ₁=−10°+/−5°,

[0056] a quartz plate, providing a retardation δ_(Q)=90°+/−5° at 633 nm,set at an orientation angle θ_(Q)=5°+/−5°,

[0057] a second ferroelectric liquid crystal, with a retardationδ₂=180°+/−15° at 510 nm, set at an orientation angle θ₂=71°+/−10°,

[0058] This polarimetric system is most conveniently coupled with aspectrometer placed after the analyzer of the PSD, and equipped with amultipoint detector (typically a CCD), which allows the polarimetricanalysis to be carried out simultaneously in the whole spectral rangedefined by the currently available FLC elements, and which might beextended in the future with new FLC materials.

[0059] Both embodiments described up to now are provided forillustrative purposes only and should not be used to unduly limit thescope of the present invention. For example, the polarimetric system isnot limited to the use of NLCs or FLCs in the PSG and the PSD but avariety of devices combining ferroelectric and nematic liquid crystalscan also be designed for simultaneous optimization of the conditionnumbers of the PSG and the PSD.

[0060] Said polarimetric systems are ellipsometers,

[0061] Said polarimetric systems are Mueller polarimetric systems foranalyzing a sample represented by the sixteen coefficients of a Muellermatrix,

[0062] The light beam emitted by the excitation section is in thespectral range 400-1500 nm for nematic liquid crystals and 420-800 nmfor ferroelectric liquid crystals currently available,

[0063] This spectral range might be extended in the UV or further intothe IR with new LC materials with the different embodiments describedwithin the scope of the present invention,

[0064] The excitation section comprises a monochromator positionedbefore the polarizer of the PSG,

[0065] The detection means comprises either a single detector, or amultipoint photosensitive detector, adapted with the processing unit topolarimetric imaging,

[0066] The multipoint photosensitive detector is a charge coupleddetector (CCD),

[0067] For spectroscopic applications, the detection means may comprisea spectrometer, placed after the analyzer of the PSD, and preferablycoupled with a CCD, to achieve polarimetric analysis simultaneously overthe entire spectral range,

[0068] The device can be used both in transmission and in reflectionmodes.

[0069] The invention also concerns a calibration process of apolarimetric system involving the measurement of at least a referencesample in which

[0070] one illuminates the sample with a polarized incident light beamemitted by a polarisation state generator (PSG) containing a polarizer,said PSG modulating the light beam polarization,

[0071] said sample transmits or reflects a measurement beam,

[0072] one detects the measurement beam with an analysis sectioncomprising a polarization state detector (PSD) containing an analyzerand detection means, and

[0073] one processes the electrical signals produced by the detectionmeans with a processing unit.

[0074] According to the invention,

[0075] Said PSG contains a first and a second liquid crystal elementsLC_(j) (j=1, 2) positioned after the polarizer, said LC_(j) elementshaving retardations δ_(j) between their ordinary and extraordinary axesand said extraordinary axes making angles θ_(j) with respect to thepolarization direction defined by the linear polarizer so that byvarying the retardation δ_(j) of each LC_(j) element for a fixed valueof the θ_(j) angle, when the LCj elements are nematic LCs, or byswitching the orientation angle θ_(j) when the LCj elements areferroelectric LCs, one modulates the incident light beam polarization,the PSG having a modulation matrix (W) that is non singular,

[0076] Said PSD contains a third and a fourth liquid crystal elementsLC′_(j) (j=1, 2) positioned before the analyser, said LC′_(j) elementsbeing the same as the LC_(j) elements of the PSG but positioned in thereverse order, so that by varying the retardation δ′_(j) of each elementfor fixed values of θ′_(j) angles LC′_(j) when the LC′_(i) are nematicLCs, or by switching the values of angles θ′_(j) at fixed δ′_(j) whenthe LC′_(i) are ferroelectric LCs, one generates a detection matrix (A)for the analysis section, said matrix being non singular and so that fora given set of retardations (δ_(j), δ′_(j)) (j=1, 2), or for a given setof orientation angles (θ_(j), θ′_(j)), one produces a measured quantity(D_(n)) and so that the processing unit produces the raw data matrixB=AMW, where (M) is the Mueller matrix of the sample,

[0077] The processing unit produces after n=16 of such measurements andsuitable data treatment:

[0078] in a simplified (ellipsometric) operation mode: the classicalellipsometric angles (Ψ, Δ) as well as the overall transmission (orreflection) coefficient τ characterizing the samples opticallyequivalent to DR, such as isotropic non depolarizing surfaces measuredin reflection. The measurement procedure includes a check of thevalidity of the description of the sample as a DR without the need ofmoving any part of the system, while with usual ellipsometers this canbe checked only by rotating the analysis arm by 90°,

[0079] in a complete (Mueller polarimetric) operation mode: the completeMueller matrix (M) of any sample, with its sixteen coefficients,

[0080] Said calibration processes thus comprise:

[0081] for the simplified (ellipsometric) operation mode:

[0082] for ellipsometric measurements in transmission of samples assumedto be dichroïc retarders (DR), taking a complete measurement of areference sample consisting of a DR defined by a Mueller matrix (M₀)with known parameters τ₀, Ψ₀ and Δ₀, said reference sample beingpropagation in air and (M₀) then being the identity matrix (I_(o)), saidmeasurement providing a reference raw data matrix B₀=AM₀W,

[0083] for ellipsometric measurements in reflection of samples assumedto be dichroïc retarders (DR), taking a complete measurement of areference sample consisting of a DR defined by a Mueller matrix (M_(o))with known parameters (τ_(o), Ψ_(o), Δ_(o)), said sample being ametallic mirror or a known sample for a system working in reflectionmode (such as a NIST sample made of silicon covered by a known thicknessof oxide), said measurement providing a reference raw data matrixB_(o)=AM_(o)W,

[0084] a) For the complete (Mueller polarimetric) operation mode of asystem working in transmission:

[0085] choosing a set of reference samples elements (p) comprisingdichroïc retarders with approximately known Mueller matrices (M_(p)),defined by the parameters (τ_(p),Ψ_(p),Δ_(p)) one of these elementsbeing the identity matrix (I_(o)) describing propagation in air,

[0086] for each of the reference samples (p), taking a completemeasurement of said sample, set at an orientation angle θ_(p), bymodulating the incoming light polarization and analyzing the outcominglight polarization, constructing the matrix (AR(−θ_(p))M_(p)R(θ_(p))W)using the processing unit, this matrix being a product of the detectionmatrix (A), the Mueller matrix (R(−θ_(p))M_(p)R(θ_(p))) of said elementp set at the angle θ_(p), with R(θ) a matrix describing a rotation by anangle θ about the z axis and the modulation matrix (W),

[0087] calculating the product (AI_(o)W)⁻¹(AR(−θ_(p))M_(p)R(θ_(p))W) foreach reference sample p in order to obtain an experimental matrix(C_(p)) and determining M_(p), or, more precisely, the values of itsparameters τ_(p), Ψ_(p) and Δ_(p), independently of the angles θ_(p)through the eigenvalues of C_(p), which are identical to those of M_(p).This allows a very accurate characterization of each sample in situ,during the calibration itself,

[0088] constructing a matrix (K_(tot)(θ_(p))) equal to$\sum\limits_{p}( {{H_{p}( \theta_{p} )}^{T}{H_{p}( \theta_{p} )}} )$

[0089]  where the matrix H_(p)(θ_(p)) is defined asH_(p)(θ_(p))[X]=R(−θ_(p))M_(p)R(θ_(p))X−XC_(p) where (X) is any real 4×4matrix,

[0090] determining the eigenvalues λ_(i) (i=1 to 16) of the(K_(tot)(θ_(p))) matrix in order to extract the modulation matrix (W)that verifies K_(tot)(W)=0, the p reference samples being chosen so thatone and only one eigenvalue λ_(i) vanishes when the angles (θ_(p)) usedin the calculation of K_(tot)(θ_(p)) are set equal to their actualvalues during the calibration measurements, while the other eigenvaluesλ_(j), being sorted in decreasing order of value, verify Z=λ₁₅/λ₁<1 andthe ratio Z is maximised,

[0091] This is equivalent to determine the modulation matrix W,(together will all the angles θ_(p)), as the unique solution of the setof matrix equations:

M _(p)(θ_(p))X−XC _(p)  (8)

[0092] determining the detection matrix (A) by constructing the product(AI_(o)W)(W⁻¹).

[0093] According to various embodiments, the present invention alsoconcerns the characteristics below, considered individually or in alltheir technical possible combinations.

[0094] a set of reference samples comprises

[0095] a linear polarizer set at θ₁=0° orientation,

[0096] a linear polarizer set at θ₂=90°+/−5° orientation,

[0097] a retardation plate with a retardation δ=110°+/−30° set atθ₃=30°+/−5°,

[0098] for spectroscopic applications, the retardation plate is anachromatic quarterwave plate.

[0099] b) For the complete (Mueller polarimetric) operation mode of asystem working in reflection:

[0100] choosing a set of reference samples comprising a linearpolarizer, defined by its Mueller matrix M_(pol), and a first DR1 and asecond DR2 dichroïc retarders, said DR_(i) having Mueller matrices(M_(i)), with i=(1, 2) respectively, with approximately known values ofthe parameters τ_(i), Ψ_(i), Δ_(i),

[0101] with each of the following sequence of elements, taking ameasurement by modulating the incoming light polarization and analyzingthe outcoming light polarization, the origin of the azimuthal angles(θ=0) being taken in the plane of incidence,

[0102] DR₁ alone, set at θ=0, yielding a measured matrix B₁=AM₁W

[0103] DR₂ alone, set at θ=0, yielding a measured matrix B₂=AM₂W

[0104] DR₁, set at θ=0, and preceded by the polarizer set at anorientation angle θ₁, yielding a measured matrixB_(p1)=AM₁R(−θ₁)M_(pol)R(θ₁)W, where R(θ) is a matrix describing arotation by an angle θ about the z axis

[0105] DR₁, set at θ=0, and followed by the polarizer, set at anorientation angle θ₂, yielding the measured matrixB_(p2)=AR(−θ₂)M_(pol)R(θ₂)M₁W,

[0106] Calculating the products C₁=B₂ ⁻¹B₁ and C₂=B₁B₂ ⁻¹ and then thematrices N₁=M₂ ⁻¹M₁ and N₂=M₁M₂ ⁻¹ through their eigenvalues, which arethe same as those of C₁ and C₂.

[0107] N₁ and N₂ have actually the form of the Mueller matrices of DR,which are oriented, by definition, at θ=0.

[0108] Calculating the products C_(p1)=B₂ ⁻¹B_(p1)=W⁻¹N₁R(−θ₁)M_(pol)R(θ₁)W and C_(p2)=B_(p2)B₂⁻¹=AR(−θ₂)M_(pol)R(θ₂)N₂A⁻¹.

[0109] Constructing a 16×16 real matrix K₁(θ₁) as K₁(θ₁)=H₁^(T)H₁+H_(p1)(θ₁)^(T)H_(p1)(θ₁), where, for any real 4×4 real matrix X,H₁[X] and H_(p1)(θ₁)[X] are defined as H₁[X]=N₁X−XC₁ andH_(p1)(θ₁)[X]=N₁M_(pol)(θ₁)X−XC_(p1),

[0110] Determining the modulation matrix W and the orientation angle θ₁by requiring that K₁(θ₁) has one vanishing eigenvalue, and W is thevector associated with this vanishing eigenvalue,

[0111] Constructing a 16×16 real matrix K₂(θ₂) as K₂(θ₂)=H₂^(T)H₂+H_(p2)(θ₂)^(T)H_(p2)(θ₂), where, for any real 4×4 real matrix X,H₂[X] and H_(p2)(θ₂)[X] are defined as H₂[X]=C₂X−XN₂ andH_(p2)(θ₂)[X]=C_(p2)X−XM_(pol)(θ₂)N₂

[0112] Determining the analysis matrix A and the orientation angle θ₂ byrequiring that K₂(θ₂) has one vanishing eigenvalue, and A is the vectorassociated with this vanishing eigenvalue,

[0113] reference samples are then chosen according to the followingcriteria:

[0114] the 16×16 real symmetrical matrices K₁(θ₁) and K₂(θ₂) will onlyhave one vanishing eigenvalue, if and only if the angles θ₁ and θ₂ usedfor their evaluation are equal to the azimuthal angles of the polarizersduring the calibration measurements,

[0115] The next eigenvalues are as large as possible, or, moreprecisely, the ratios Z=λ₁₅/λ₁ of the smallest nonvanishing eigenvalues(λ₁₅) over the largest (λ₁) eigenvalues of K₁ and K₂ are as large aspossible.

[0116] According to various embodiments, the present invention alsoconcerns the characteristics below, considered individually or in alltheir technical possible combinations.

[0117] A set of reference samples is

[0118] a linear polarizer set at θ₁=45°+/−5°,

[0119] a linear polarizer set at θ₂=−45°+/−5°, and

[0120] a couple of samples equivalent to a first DR, and a second DR₂dichroïc retarders, both oriented at θ=0 with respect to the incidenceplane, with Mueller matrices M₁ and M₂ such that the products M₂ ⁻¹M₁and M₂ ⁻¹M₁ are the Mueller matrices of a DR with Ψ=45°+/−30° andΔ=90°+/−10,

[0121] For spectroscopic applications, said reference samples comprise ametallic mirror,

[0122] For spectroscopic applications said reference samples comprise anachromatic quarter-wave plate, oriented with one axis in the incidenceplane placed before or after a metallic mirror.

[0123] The invention also regards two measurement processes providing,after suitable instrument calibration

[0124] in the ellipsometric mode, the parameters (τ,Ψ,Δ) of a sampleassumed to be a dichroïc retarder (DR),

[0125] in the complete polarimetric mode, the Mueller matrix (M) of anysample

[0126] Both measurement processes involve, in all cases:

[0127] emitting an incident light beam linearly polarised along adirection of polarisation (i),

[0128] modulating the incident beam polarization,

[0129] sending the modulated incident beam to the sample, and returninga measurement beam,

[0130] collecting the measurement beam through a polarisation analysissection,

[0131] detecting the measurement beam after the polarization analysissection and producing electrical signals forming the raw data matrixB=AMW,

[0132] transmitting the electrical signals to a processing unit,

[0133] According to the invention,

[0134] modulating the incident beam polarization by means of two liquidcrystal elements LC_(j) (j=1, 2) by varying either the angularorientations θ_(j), of the extraordinary axes with respect to thepolarization direction (i) of the linear polarizer when the liquidcrystals (LCs) comprise ferroelectric LCs (FLCs), or the retardationsδ_(j) at fixed orientations when the LCs comprise nematic LCs (NLCs),

[0135] producing measured quantities (D_(n)) by means of an analysissection comprising two liquid crystal elements LC′_(j) (j=1, 2) byvarying the retardation δ′_(j) of each element for fixed values ofθ′_(j) angles when the LCs are NLCs, or the values of the orientationangles θ′_(j) for fixed values of retardation δ′_(j) (j=1, 2) when theLCs are FLCs,

[0136] The raw data B are then processed as follows:

[0137] in the ellipsometric mode:

[0138] calculating the matrix C=B₀ ⁻¹ B=W⁻¹ M₀ ⁻¹ M W, where B₀=AM₀W isthe raw data matrix taken with the calibration sample. The eigenvaluesof C are the same as those of M₀ ⁻¹ M, which has the same form as theMueller matrix of a DR. As a result, two of these eigenvalues (λ_(R1)and λ_(R2)) are positive real, while the other two (λ_(c1) and λ_(c2))are complex conjugates.

[0139] deducing the ellipsometric parameters (τ, Ψ, Δ) of the studiedsample from these eigenvalues and the known parameters τ₀, Ψ₀ and Δ₀ ofthe reference sample according to: $\begin{matrix}{\Psi = {\arctan ( {\sqrt{\frac{\lambda_{R1}}{\lambda_{R2}}}\tan \quad \Psi_{0}} )}} & (4) \\{\Delta = {\Delta_{0} + {\frac{1}{2}A\quad r\quad {g( \frac{\lambda_{C1}}{\lambda_{C2}} )}}}} & (5) \\{\tau = \frac{{\tau_{0}( {\lambda_{R1} + \lambda_{R2}} )}\sin^{2}2\quad \Psi_{0}}{( {1 - {\cos \quad 2\Psi_{0}\cos \quad 2\Psi}} )}} & (6)\end{matrix}$

[0140] checking the validity of the description of the sample as a DRfrom the following relationship

|λ_(C1)|²=|λ_(C2)|²=λ_(R1)λ_(R2)  (7)

[0141]  which must be obeyed by the eigenvalues of any Mueller matrixdescribing a DR, as it can be directly seen from eq. (3).

[0142] in the complete polarimetric mode:

[0143] the Mueller matrix M of any sample is calculated from the rawdata matrix B as M=A⁻¹BW⁻¹.

[0144] To facilitate further the description of the invention, thefollowing drawings are provided in which:

[0145]FIG. 1 is a schematic view of a polarimetric system operated intransmission according to the invention.

[0146]FIG. 2 is a schematic view of a spectroscopic ellipsometric systemoperated in reflection according to the invention.

[0147]FIG. 3 shows the experimental values of the diagonal blockelements of the Mueller matrix of a high quality Babinet-Soleilcompensator as a function of the setting x (mm) of the micrometric screwcontrolling the compensator retardation Δ (FIG. 3a), together with aplot of the values of Δ (deduced from the measured matrix elementsaccording to eqs.(2)) versus the setting x (mm) of the micrometric screw(squares) and the corresponding linear regression (solid line) (FIG.3b).

[0148]FIG. 4 shows the off diagonal block elements of the same Muellermatrix versus the setting x (mm) of the micrometric screw controllingthe compensator retardation Δ.

[0149] These drawings are provided for illustrative purposes only andshould not be used to unduly limit the scope of the invention.

[0150]FIG. 1 shows a polarimetric system according to an embodiment ofthe invention. It contains an excitation section 1 emitting a light beam2 and an analysis section 3.

[0151] The excitation section 1 comprises a polarization state generator4 (PSG) through which passes the light beam 2. The polarization stategenerator 4 comprises a polarizer 5 that linearly polarizes the lightbeam 2 along a polarization direction (i).

[0152] First optical means 6 defines the geometry of the beam 2 at thesample 7.

[0153] The analysis section 3 comprises a polarization state detector 8(PSD) containing an analyzer 9 and detection means 10 for detecting thelight beam 2.

[0154] In a particular embodiment, the detection means 10 comprises amultipoint photosensitive detector that produces electrical signals sentto a processing unit 11. Said detection means 10 are adapted topolarimetric imaging and the multipoint photosensitive detector isadvantageously a charge coupled detector (CCD).

[0155] The polarimetric system may comprise as well a monochromatorwhich is located in a first embodiment within the light source 12 thatemits the light beam 2, before said beam enters the polarization stategenerator 4. In a second embodiment, the monochromator is located withinthe detection means 10, after the light beam exits the polarizationstate detector 8.

[0156] According to the invention, the polarization state generator 4 ofthe polarimetric system comprises a first and a second liquid crystalelements 13 having birefringent axes, said liquid crystal elements 13being positioned after the polarizer 5. The polarization state detector8 also comprises a first and a second liquid crystal elements 14 havingbirefringent axes and positioned before the analyzer 9. The polarimetricsystem comprises also control means for controlling said liquid crystalelements 13, 14.

[0157] The present invention concerns Mueller polarimetric systems foranalyzing samples represented by the sixteen coefficients of a Muellermatrix. The polarization state generator (PSG) 4 and the polarisationstate detector (PSD) 8 comprise each a first and a second liquid crystalelements 13, 14 LC_(j) (j=1, 2) which may either be nematic liquidcrystals (NLC) or ferroelectric liquid crystals (FLC).

[0158] When NLCs are used, each NLC_(j) element 13 of the PSG 4(respectively, for each NLC_(j) element 14 of the PSD 8), has anextraordinary axis making a fixed angle θ_(j) (resp. θ′_(j)) withrespect to the direction of polarisation (i) and a variable retardationδ_(j) (resp. δ′_(j)) between its ordinary and extraordinary axes, whichcan be controlled electronically, said liquid crystal (NLC_(j)) elements14 being positioned in reverse order in the PSD 8 with respect to theNLC_(j) elements 13 of the PSG 4.

[0159] When FLCs are used, each FLC_(j) element 13 of the PSG 4(respectively, for each FLC_(j) 14 element of the PSD 8), has a constantretardation δ_(j) (resp. δ′_(j)) between its ordinary and extraordinaryaxes, and the angle θ_(j) (resp. θ′_(j)) between the extraordinary axisof the FLC and the direction of polarization (i) can be switched betweentwo values separated by 45° by means of an electronic control device.Said liquid crystal (FLC_(j)) elements 14 are positioned in reverseorder in the PSD 8 with respect to the FLC_(j) elements 13 of the PSG 4.

[0160] It is known that the application by control means of anappropriate voltage signal on each liquid crystal element (FLC or NLC)13, 14 allows modulating the polarization of a light beam passingthrough said liquid crystal elements 13, 14.

[0161] In mathematical terms, the liquid crystal elements LC_(j) 13 ofthe polarization state generator 4 applies a polarization modulationsuch that the Stokes vector (S) of the light beam 2 at the exit of thepolarization state generator 4 is given by: $\begin{matrix}{S = {( D^{\delta_{2}\theta_{2}} )\quad ( D^{\delta_{1}\theta_{1}} )\quad \begin{pmatrix}\begin{matrix}\begin{matrix}1 \\1\end{matrix} \\0\end{matrix} \\0\end{pmatrix}}} & (8)\end{matrix}$

[0162] where D^(δ) ^(_(j)) ^(θ) ^(_(j)) is the Mueller matrix of theLC_(j) element (j=1,2). When a set of four couples of retardations(δ₁,δ₂) or angles (θ₁,θ₂) are defined sequentially by the control meansof the LCs, four linearly independent Stokes vectors are hence generatedfrom an unpolarized light source.

[0163] In a preferred embodiment, the liquid crystal (LC) elements 13according to the invention are nematic liquid crystal cells (NLC). Saidliquid cells are particularly suitable to polarimetric imaging sincetheir typical transmission range is currently between 400 nm and 1500nm, and could be extended with new liquid crystal materials.

[0164] With NLCs and in a preferred embodiment, the orientation anglesθ′_(j) (j=1,2) are equal to θ_(j) (j=1,2) and the retardations δ′_(j)(j=1,2) are equal to −δ_(j) (j=1,2) (modulo 2π). Advantageously then,the couple of retardations (δ₁, δ₂) takes sequentially the followingvalues: (Δ₁,Δ₁), (Δ₁,Δ₂), (Δ₂,Δ₁), (Δ₂,Δ₂), where Δ₁ and Δ₂ verify theformulae (315°+p 90°) and (135°+p 90°) respectively, where p is the sameinteger in both formulae and the angles θ₁ and θ₂ verify the formulae (ε27°+q 90°) and (ε 72°+r 90°) respectively where ε=±1 has the same valuein both equations while q and r are any integer, with tolerances on theangles θ_(i) and on the retardations Δ_(i) equal to +−10° and +−20°respectively. This embodiment allows the simultaneous optimization ofthe condition numbers s(W) and s(A), both A and W matrices being nonsingular.

[0165] The LC elements 13, 14 may also be ferroelectric liquid crystalmodulators or any other suitable liquid crystal light modulating device.

[0166] With FLCs and in a preferred embodiment, the extraordinary axesof the FLC make a couple of azimuthal angles (θ₁,θ₂) (resp. (θ′₁,θ′₂))with respect to the direction of the input polarizer 5 of the PSG 4(resp. the output analyzer 9 of the PSD, and by means of a suitableelectronic control device, these angles are set sequentially to (θ₁,θ₂),(θ₁+45°,θ₂), (θ₁,θ₂+45°), (θ₁+45°,θ₂+45°). The azimuthal angles θ′_(j)are equal to θ_(j) (j=1,2) and the retardations δ′_(j) are equal to−δ_(j) (j=1,2) (modulo 2π) for simultaneous optimization of thecondition numbers s(W) and s(A) of the modulation and analysis matrices.

[0167] The retardations δ_(i) are given by δ₁=80°+−15° and δ₂=160°+−15°,while the orientation angles θ_(i) are given by θ₁=67°+−10° andθ₂=160°+−40°.

[0168] With FLCs and for spectroscopic applications (i.e. operation atvariable wavelengths), as the values of the retardations (δ₁, δ₂) arenot electrically controllable as for nematic liquid crystals, abirefringent plate is preferentially inserted between the two FLCselements, leading to an overall optimization of A and W matrices in thewhole transparency range of the FLCs, which is currently from 420 nm to800 nm, and might be extended in the future with new materials. Withtypical values of FLC birefringence dispersion, the retardations (δ₁,δ₂) can be advantageously chosen equal to (90°, 180°) in the green partof the spectrum, while the birefringent plate can be chosen as a zeroorder quarter wave in the red part of the spectrum (633 nm), said platebeing made of quartz. Hence, in a particular embodiment and forspectroscopic applications, the birefringent plate is a quartz plate andthe PSG 4 comprises:

[0169] a linear polarizer, set by definition at an orientation angleθ=0,

[0170] a first ferroelectric liquid crystal, with a retardationδ₁=90°+−5° at 510 nm, set at an orientation angle θ₁=−10°+−5°,

[0171] a quartz plate, providing a (true zero order) retardationδ_(Q)=90°+−5° at 633 nm, set at an orientation angle θQ=5°+−5°,

[0172] a second ferroelectric liquid crystal, with a retardationδ₂=180°+−15° at 510 nm, set at an orientation angle θ₂=71°+−10°.

[0173] The invention regards as well conventional ellipsometry that is aspecial case of polarimetry for isotropic layers with smooth interfaces.

[0174]FIG. 2 shows a particular embodiment in which a spectroscopicellipsometric system based on ferroelectric liquid crystals (FLCs)contains an excitation section 1 emitting a light beam 2, a sampleholder 15 and an analysis section 3.

[0175] The excitation section 1 comprises a polarization state generator4 (PSG) through which passes the light beam 2. The polarization stategenerator 4 comprises a polarizer 5 that linearly polarizes the lightbeam 2 along a polarization direction (i). First optical means 6 focusesthe beam 2 at the sample 7.

[0176] The incidence angle of the light beam 2 on the sample surface isdefined as the angle at which the focused beam strikes the samplesurface with respect to the normal to the surface 7. For example, a beam2 with normal incidence at the sample surface has an incidence angle ofzero degree. The angle of incidence of the beam can be advantageouslyvaried. The purpose of the focusing beam is to obtain a small spot onthe sample 7, i.e. a compact spot with preferably dimensions inferior toa few tenths of mm². This spot should provide a lateral resolutionsufficient to map the sample surface. The light beam 2 emitted by theexcitation section 1 is in the transparency spectral region of the FLC,which is currently from 420 to 800 nm, and might be extended furtherwith new FLC materials.

[0177] The beam 2 reflects off the sample surface and passes through theanalysis section 3. In a more general case, the beam is scattered by thesample surface and passes through the analysis section 3. The analysissection 3 comprises an input optical (collimating) system 16, apolarisation state detector 8 (PSD) containing an analyzer 9 anddetection means 10 for detecting the light beam 2. The detection means10 typically comprises a spectrometer coupled to several photodetectors,typically an array of CCD (charge coupled devices) that produceselectrical signals. A processing unit 11 receives said electricalsignals.

[0178] According to the invention, the polarization state generator 4 ofthe polarimetric system comprises a first and a second ferroelectricliquid crystal elements 13 having birefringent axes, said liquid crystalelements 13 being positioned after the polarizer 5, and a fixedretardation plate 17, between the two liquid crystal elements 13. Thepolarization state detector 8 also comprises a first and a secondferroelectric liquid crystal elements 14 having birefringent axes andpositioned before the analyzer 9, and a fixed retardation plate 18 setbetween the FLCs. The polarimetric system comprises also control meansfor controlling said liquid crystal elements 13, 14.

[0179] The present invention can also be advantageously implemented forpolarimetric imaging by a CCD camera.

[0180] The polarimetric system and the polarimetric measurement processaccording to the invention have been the object of variousimplementations whose following example demonstrates the quality of theresults obtained.

[0181]FIGS. 3 and 4 show the results obtained with a polarimetric systembased on nematic liquid crystals, calibrated and operated intransmission, at 633 nm. The test sample was a high qualityBabinet-Soleil compensator, which can be considered as dichroïc retarder(DR) with Ψ≈45° and a retardation Δ which is a linear function of thesetting of the compensator micrometric screw. The 16 elements of thecompensator Mueller matrix (M) were measured for different settings ofthis screw.

[0182] The experimental values for the diagonal block elements are shownon FIG. 3a, versus the screw settings (in mm) of the micrometric screwcontrolling the compensator retardation Δ; these values follow quiteclosely (to within 0.01) the behavior expected from eqs.(2) for Ψ=45°.In FIG. 3b is shown the variation of the dephasing Δ, as deduced fromthe values of the lower diagonal block elements, as a function of thescrew setting: the standard deviation from a perfect linear fit is0.13°, equivalent to λ/2700, which is even better than the accuracyspecified by the manufacturer of the compensator, thus confirming theperformance of this polarimetric technique.

[0183]FIG. 4 shows the results obtained for the off-diagonal blockelements of the Mueller matrix, again as a function of the micrometricscrew setting (in mm). We recall that these elements are expected tovanish. Some of these elements, shown on FIG. 4a, are independent of thecompensator orientation, and they are always found to be smaller than5.10⁻³ in absolute value. For the other ones (FIG. 4b) this absolutevalue can reach 1.5 10⁻². This latter value might be due in part to animperfect alignment of the components within the compensator itself.

1. A polarimetric system for analyzing a sample comprising: anexcitation section (1) emitting a light beam, said excitation section(1) comprising a polarization state generator (4) containing a polarizer(5) linearly polarizing the incident light beam (2) along a direction ofpolarization (i), an analysis section (3) comprising a polarizationstate detector (8) containing an analyzer (9), and detection means (10),a processing unit (11), wherein, the polarization state generator (4)(PSG) and the polarization state detector (8) (PSD) comprise each afirst and a second liquid crystal elements (13, 14) LC_(j) (j=1, 2)having, for each LC_(j) element (13) of the PSG (respectively for eachLC′_(j) element (14) of the PSD), an extraordinary axis making an angleθ_(j) (resp. θ′_(j)) with respect to the direction of polarization (i)and a retardation δ_(j) (resp. δ′_(j)) between its ordinary andextraordinary axes, said liquid crystal (LC′_(j)) elements (14) beingpositioned in reverse order in the PSD with respect to the LC_(j)elements (13) of the PSG, and the orientation angles θ′_(j) are equal toθ_(j) (j=1,2) and the retardations δ′_(j) are equal to −δ_(j) (j=1,2),(modulo 2π).
 2. A polarimetric system according to claim 1, wherein saidliquid crystal elements (13, 14) LC_(j) (j=1, 2) are nematic (NLCs)liquid crystals and the polarimetric system comprises an electroniccontrol unit enabling polarization modulation by varying theretardations δ_(j) and δ′_(j) for NLCs.
 3. A polarimetric systemaccording to claim 1 wherein said liquid crystal elements (13, 14)LC_(j) (j=1,2) are ferroelectric (FLCs) liquid crystals, and thepolarimetric system comprises an electronic control enablingpolarization modulation by varying the orientation angles θ_(j) andθ′_(j) for FLCs.
 4. A polarimetric system according to claims 1 and 2,wherein: the couple of retardations (δ₁, δ₂) is varied in the followingsequence (Δ₁,Δ₁), (Δ₁,Δ₂), (Δ₂,Δ₁), (Δ₂,Δ₂), where Δ₁ and Δ₂ verify theformulae (Δ₁=315°+p 90°) and (Δ₂=135°+p 90°) respectively, where p isthe same integer in both formulae, with a tolerance of +/−20°, theorientations angles θ₁ and θ₂ verify the formulae (θ₁=ε 27°+q 90°) and(θ₂=ε 72°+r 90°) respectively where ε=±1 has the same value in bothequations while q and r are any integer, with a tolerance of +/−10°. 5.A polarimetric system according to claims 1 and 3, wherein: theorientations of the extraordinary axes are set sequentially to (θ₁,θ₂),(θ₁+45°,θ₂), (θ₁,θ₂+45°), (θ₁+45°,θ₂+45°), the retardations (δ₁, δ₂)verify δ₁=80°+/−15° and δ₂=160°+/−15°, while the orientation angles (θ₁,θ₂) are given by θ₁=67°+/−10° and θ₂=160°+/−40°.
 6. A polarimetricsystem according to claim 5, wherein the polarimetric system is suitablefor a range of wavelengths, and a fixed retardation plate (17, 18) isplaced between said two FLCs, both in the PSG and in the PSD.
 7. Apolarimetric system according to claim 6, wherein said polarimetricsystem is optimized for the spectral range from 420 nm to 800 nm, andthe retardation plate is a quartz plate and PSG comprises a linearpolarizer, set at an orientation angle θ=0, a first ferroelectric liquidcrystal, with a retardation δ₁=90°+−5° at 510 nm, set at an orientationangle θ₁=−10°+/−5°, a quartz plate, providing a retardationδ_(Q)=90°+/−5° at 633 nm, set at an orientation angle θ_(Q)=5°+/−5°, asecond ferroelectric liquid crystal, with a retardation δ₂=180°+/−15° at510 nm, set at an orientation angle θ₂=71°+/−10°.
 8. A polarimetricsystem according to claim 1 to 7 wherein said polarimetric system is anellipsometer.
 9. A polarimetric system according to any one of theclaims 1 to 7, wherein said polarimetric system is a Muellerpolarimetric system for analyzing a sample (7) through the measurementof the sixteen coefficients of its Mueller matrix.
 10. A polarimetricsystem according to any one of claims 1 to 9 wherein the light beam (2)emitted by the excitation section (1) is in the spectral range 400-1500nm for nematic liquid crystals, and 420-800 nm for ferroelectric liquidcrystals. 11 A polarimetric system according to any one of claims 1 to10 wherein the excitation section (1) comprises a monochromatorpositioned before the polarization state generator (4) (PSG).
 12. Apolarimetric system according to any one of claims 1 to wherein thedetection means (10) comprises a monochromator, placed after the PSD.13. A polarimetric system according to any one of claims 1 to 12 whereinthe detection means (10) comprises a multipoint photosensitive detector,adapted with the processing unit (11) to polarimetric imaging.
 14. Apolarimetric system according to claim 13 wherein the multipointphotosensitive detector is a charge coupled detector (CCD).
 15. Acalibration process of a polarimetric system involving measurement of atleast a reference sample (7) in which one illuminates the sample (7)with a polarized incident light beam (2) emitted by a polarisation stategenerator (PSG) containing a polarizer, said PSG modulating the lightbeam (2) polarization, said sample (7) transmits or reflects ameasurement beam, one detects the measurement beam with an analysissection (3) comprising a polarization state detector (8) (PSD)containing an analyzer (9), and detection means (10), and one processesthe electrical signals produced by the detection means (10) with aprocessing unit (11), wherein, Said PSG contains a first and a secondliquid crystal elements (13) LC_(j) (j=1,2) positioned after thepolarizer, said LC_(j) elements (13) having retardations δ_(j) betweentheir ordinary and extraordinary axes and said extraordinary axes makingangles δ_(j) with respect to the polarization direction defined by thelinear polarizer so that by varying the retardation δ_(j) of each LC_(j)element for a fixed value of the θ_(j) angle, when the LC_(j) elementsare nematic LCs, or by switching the orientation angle θ_(j) when theLC_(j) elements are ferroelectric LCs, one modulates the incident lightbeam (2) polarization, the PSG having a modulation matrix (W) that isnon singular, Said PSD contains a third and a fourth liquid crystalelements (14) LC′_(j) (j=1,2) positioned before the analyser, saidLC′_(j) elements (14) being the same as the LC_(j) elements of the PSGbut positioned in the reverse order, so that by varying the retardationδ′_(j) of each element for fixed values of θ′_(j) angles when theLC′_(j) are nematic LCs, or by switching the values of angles θ′_(j) forfixed δ′_(j) when the LC′_(j) are ferroelectric LCs, one generates adetection matrix (A) for the analysis section (3), said matrix being nonsingular and so that for a given set of retardations (δ_(j), δ′_(j))(j=1,2), or for a given set of orientation angles (θ_(j), θ′_(j)), oneproduces a measured quantity (D_(n)) and so that the processing unit(11) produces the raw data matrix B=AMW, where (M) is the Mueller matrixof the sample (7).
 16. A calibration process according to claim 15, saidcalibration process being adapted to ellipsometric measurements intransmission of samples (7) assumed to be dichroïc retarders (DR), inorder to determine their ellipsometric parameters (τ, Ψ, Δ) wherein theprocess comprises taking a complete measurement of a reference sample(7) consisting of a DR defined by a Mueller matrix (M₀) with knownparameters τ₀, Ψ₀ and Δ₀, said reference sample (7) being propagation inair and (M₀) then being the identity matrix (I_(o)), said measurementproviding a reference raw data matrix B₀=AM₀W.
 17. A calibration processaccording to claim 15, said calibration process being adapted toellipsometric measurements in reflection of samples (7) assumed to bedichroïc retarders (DR) in order to determine their ellipsometricparameter (τ, Ψ, Δ) wherein the process comprises taking a completemeasurement of a reference sample (7) consisting of a DR defined by aMueller matrix (M_(o)) with known parameters (τ_(o), Ψ_(o), Δ_(O)), saidsample (7) being a metallic mirror or a known sample (7) for a systemworking in reflection mode, said measurement providing a reference rawdata matrix B₀=AM₀W.
 18. A calibration process according to claim 15,said process being adapted to the complete Mueller polarimetry of anysample (7) in transmission, wherein the calibration process comprises:choosing a set of reference samples (7) elements (p) comprising dichroïcretarders with approximately known Mueller matrices (M_(p)), defined bythe parameters (τ_(p),Ψ_(p),Δ_(p)) one of these elements being theidentity matrix (I_(o)) describing propagation in air, for each of thereference samples (p), taking a complete measurement of said sample (7),set at an orientation angle θ_(p), by modulating the incoming lightpolarization and analyzing the outcoming light polarization,constructing the matrix (AR(−θ_(p))M_(p)R(θ_(p))W) using the processingunit, this matrix being a product of the detection matrix (A), theMueller matrix (R(−θ_(p))M_(p)R(θ_(p))) of said element p set at theangle θ_(p), with R(θ) a matrix describing a rotation by an angle θabout the z axis and the modulation matrix (W), calculating the product(AI_(o)W)⁻¹(AR(−θ_(p))M_(p)R(θ_(p))W) for each reference sample (7) p inorder to obtain an experimental matrix (C_(p)), determining the actualvalues of (τ_(p),Ψ_(p),Δ_(p)), and thus the matrix M_(p), independentlyof the angles θ_(p), from the eigenvalues of (C_(p)), constructing amatrix (K_(tot)(θ_(p))) equal to$\sum\limits_{p}( {{H_{p}( \theta_{p} )}^{T}{H_{p}( \theta_{p} )}} )$

 where the matrix H_(p)(θ_(p)) is defined asH_(p)(θ_(p))[X]=R(−θ_(p))M_(p)R(θ_(p))X−XC_(p) where (X) is any real 4×4matrix, determining the eigenvalues λ_(i) (i=1 to 16) of the(K_(tot)(θ_(p))) matrix in order to extract the modulation matrix (W)that verifies K_(tot)(W)=0, the p reference samples (7) being chosen sothat one and only one eigenvalue λ_(i) vanishes when the angles (θ_(p))used in the calculation of K_(tot)(θ_(p)) are set equal to their actualvalues during the calibration measurements, while the other eigenvaluesλ_(j), being sorted in decreasing order of value, verify Z=λ₁₅/λ₁<1 andthe ratio Z is maximised, determining the angles (θ_(p)) by requiringone of the eigenvalues K_(tot)(θ_(p)) to vanish, W being the associatedeigenvector, determining the detection matrix (A) by constructing theproduct (AI_(o)W) (W⁻¹).
 19. A calibration process according to claim 18wherein a set of reference samples (7) comprises a linear polarizer setat θ₁=0° orientation, a linear polarizer set at θ₂=90°+/−5° orientation,a retardation plate with a retardation δ=110°+/−30° set at θ₃=30°+/−5°.20. A calibration process according to claim 19, wherein the retardationplate is an achromatic quarterwave plate.
 21. A calibration processaccording to claim 15, said calibration process being adapted to thecomplete Mueller polarimetry of a sample (7) in reflection: choosing aset of reference samples (7) comprising a linear polarizer, defined byits Mueller matrix M_(pol), and a first DR1 and a second DR2 dichroïcretarders, said DR1 having Mueller matrices (M_(i)), with i=(1, 2)respectively, with approximately known values of the parameters τ_(i),Ψ_(i), Δ_(i,) with each of the following sequence of elements, taking ameasurement by modulating the incoming light polarization and analyzingthe outcoming light polarization, the origin of the azimuthal angles(θ=0) being taken in the plane of incidence, DR₁ alone, set at θ=0,yielding a measured matrix B₁=AM₁W DR₂ alone, set at θ=0, yielding ameasured matrix B₂=AM₂W DR₁, set at θ=0, and preceded by the polarizer(5) set at an orientation angle θ₁, yielding a measured matrixB_(p1)=AM₁R(−θ₁)M_(pol)R(θ₁)W, where R(θ) is a matrix describing arotation by an angle θ about the z axis DR₁, set at θ=0, and followed bythe polarizer, set at an orientation angle θ₂, yielding the measuredmatrix B_(p2)=AR(−θ₂)M_(pol)R(θ₂)M₁W, Calculating the products C₁=B₂⁻¹B₁ and C₂=B₁B₂ ⁻¹ and then the matrices N₁=M₂ ⁻¹M₁ and N₂=M₁M₂ ⁻¹through their eigenvalues, which are the same as those of C₁ and C₂,Calculating the products C_(p1)=B₂ ⁻¹ B_(p1)=W⁻¹N₁R(−θ₁)M_(pol)R(θ₁)Wand C_(p2)=B_(p2)B₂ ⁻¹=AR(−θ₂)M_(pol)R(θ₂)N₂A⁻¹, Defining a K₁ matrix asK₁(θ₁)[X]=H₁ ^(T)H₁+H_(p1)(θ₁)^(T H) _(p1)(θ₁), where for any any 4×4real matrix X, H₁[X] and H_(p1)(θ₁)[X] are defined as H₁[X]=N₁X−XC₁ andH_(p1)(θ₁)[X]=N₁R(−θ₁)M_(pol)R(θ₁)X−XC_(p1) Determining the modulationmatrix W and the orientation θ₁ by requiring that K₁(θ₁) has onevanishing eigenvalue, and W is the eigenvector associated with thisvanishing eigenvalue, Determining similarly the analysis matrix A as theeigenvector associated with the unique vanishing eigenvalue of thematrix K ₂(θ₂)[X]=H ₂ ^(T) H ₂ +H _(p2)(θ₂)^(T) H _(p2)(θ₂) where forany real 4×4 matrix X H ₂ [X]=C ₂ X−XN ₂ , H _(p2)(θ₂)[X]=C _(p2)X−XR(−θ ₂)M _(pol) R(θ₂)N₂ reference samples (7) are then chosenaccording to the following criteria: the 16×16 real symmetrical matricesK₁(θ₁) and K₂(θ₂) will only have one vanishing eigenvalue, if and onlyif the angles θ₁ and θ₂ used for their evaluation are equal to theazimuthal angles of the polarizers during the calibration measurements,The next eigenvalues are as large as possible, or, more precisely, theratios Z=λ₁₅/λ₁ of the smallest nonvanishing eigenvalues (λ₁₅) over thelargest (λ₁) eigenvalues of K₁ and K₂ are as large as possible.
 22. Acalibration process according to claim 21, wherein the set of referencesamples (7) a linear polarizer set at θ₁=45°+/−5° a linear polarizer setat θ₂=−45°+/−5°, and a couple of samples (7) equivalent to a first DR₁and a second DR₂ dichroïc retarders, both oriented at θ=0 with respectto the incidence plane, with Mueller matrices M₁ and M₂ such that theproducts M₂ ⁻¹M₁ and M₂ ⁻¹M₁ are the Mueller matrices of a DR withΨ=45°+/−30° and Δ=90°+/−10.
 23. A calibration process according to claim22, wherein for spectroscopic applications, said reference samples (7)comprise a metallic mirror.
 24. A calibration process according to claim22, wherein for spectroscopic applications said reference samples (7)comprise an achromatic quarter-wave plate, oriented with one axis in theincidence plane placed before or after a metallic mirror.
 25. Apolarimetric measurement process intended for measurement of a sample(7) represented by the coefficients of a Mueller matrix (M), in which:emitting an incident light beam (2) linearly polarised along a directionof polarisation(i), modulating the incident beam polarization, sendingthe modulated incident beam to the sample (7), and returning ameasurement beam, collecting the measurement beam through a polarisationanalysis section (3), detecting the measurement beam after thepolarization analysis section (3) and producing electrical signalsforming the raw data matrix B=AMW, transmitting the electrical signalsto a processing unit, wherein modulating the incident beam polarizationby means of two liquid crystal elements LC_(j) (j=1, 2) (13) by varyingeither the angular orientations θ_(j), of the extraordinary axes withrespect to the polarization direction (i) of the linear polarizer (5)when the liquid crystals (LCs) comprise ferroelectric LCs (FLCs), or theretardations δ_(j) at fixed orientations when the LCs comprise nematicLCs (NLCs), producing measured quantities (D_(n)) by means of ananalysis section (3) comprising two liquid crystal elements (14) LC′_(j)(j=1, 2) by varying the retardation δ′_(j) of each element for fixedvalues of θ′_(j) angles when the LCs are NLCs, or the values of theorientation angles θ′_(j) for fixed values of retardation δ′_(j) (j=1,2)when the LCs are FLCs, producing by means of the processing unit theMueller matrix (M=A⁻¹B W⁻¹) with its sixteen coefficients.
 26. Anellipsometric measurement procedure providing the ellipsometricparameters (τ,Ψ,Δ) of a sample (7) to be analysed, said sample (7) beingassumed to be a dichroïc retarder (DR), wherein the procedure comprisescollecting the raw data matrix B according to claim 25 calculating thematrix C=B₀ ⁻¹ B=W⁻¹ M₀ ⁻¹ M W, the eigenvalues of C being the same asthose of M₀ ⁻¹ M, and two of them, λ_(R1) and λ_(R2) being realpositive, while the other two λ_(C1) and λ_(C2) are complex conjugates.deducing the ellipsometric parameters of the studied sample (7) fromthese eigenvalues and the known parameters τ₀, Ψ₀ and Δ₀ of thecalibration sample (7) using $\begin{matrix}{\Psi = {\arctan ( {\sqrt{\frac{\lambda_{R1}}{\lambda_{R2}}}\tan \quad \Psi_{0}} )}} \\{\Delta = {\Delta_{0} + {\frac{1}{2}A\quad r\quad {g( \frac{\lambda_{C1}}{\lambda_{C2}} )}}}} \\{{\tau = \frac{{\tau_{0}( {\lambda_{R1} + \lambda_{R2}} )}\sin^{2}2\quad \Psi_{0}}{( {1 - {\cos \quad 2\Psi_{0}\cos \quad 2\Psi}} )}},}\end{matrix}$

testing the validity of the description of the sample (7) as a DRthrough the relationship |λ_(C1)|²=|λ_(C2)|²λ_(R1)λ_(R2) which must beverified to within the measurement accuracy if the sample (7) to beanalysed is a DR.